Chemistry - Double Mersenne number

In mathematics, a double Mersenne number is a Mersenne number of the form

$M_{M_n} = 2^{2^n-1}-1$

where n is a positive integer.

The first few double Mersenne numbers are

M_M₁ = M₁ = 1

M_M₂ = M₃ = 7

M_M₃ = M₇ = 127

M_M₄ = M₁₅ = 32767 = 7 × 31 × 151

M_M₅ = M₃₁ = 2147483647

M_M₆ = M₆₃ = 9223372036854775807 = 7² × 73 × 127 × 337 × 92737 × 649657

M_M₇ = M₁₂₇ = 7170141183460469231731687303715884105727

A double Mersenne number that is prime is called a double Mersenne prime. Since a Mersenne number M_n can be prime only if n is prime, (see Mersenne number for a proof of this), a double Mersenne number M_{M_n} can be prime only if M_n is prime. The first values of n for which M_n is prime are n = 2, 3, 5, 7, 13, 17, 19, 31. Of these, M_{M_n} is known to be prime for n = 2, 3, 5, 7; for n = 13, 17, 19, and 31, explicit factors have been found. If another double Mersenne prime is ever found, it would almost certainly be the largest known prime number.

Categories: Number theory